Linear Differential Equations as a Data Structure

Salvy, Bruno

doi:10.1007/s10208-018-09411-x

Cited by 19 publications

(11 citation statements)

References 108 publications

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“…The notion of D-finite power series was first introduced by Stanley [47] in 1980 and studied extensively in [25,52,48]. These series like algebraic numbers can be algorithmically manipulated via its defining linear differential equations [1,44]. We will study the stability problem on D-finite power series.…”

Section: Stable D-finite Power Seriesmentioning

confidence: 99%

Stability Problems in Symbolic Integration

Chen¹

2022

Preprint

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This paper aims to initialize a dynamical aspect of symbolic integration by studying stability problems in differential fields. We present some basic properties of stable elementary functions and D-finite power series that enable us to characterize three special families of stable elementary functions involving rational functions, logarithmic functions, and exponential functions. Some problems for future studies are proposed towards deeper dynamical studies in differential and difference algebra.

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Section: Stable D-finite Power Seriesmentioning

confidence: 99%

Stability Problems in Symbolic Integration

Chen¹

2022

Preprint

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“…Generating series of sequences satisfying recurrences of the form (5)-that is, by Lemma 1, formal series solutions of linear differential equations with polynomial coefficients-are called differentially finite or holonomic. We refer the reader to [22,31] for an overview of the powerful techniques available to manipulate these series and their generalizations to several variables.…”

Section: Generating Seriesmentioning

confidence: 99%

“…To the continuous solution p(x, t) corresponds a double sequence (p k i ) where k 0 is the space index and 0 i n is the time index. Taking central differences for the derivatives in (31) and letting a = (c∆t/∆x) 2 leads to (32)…”

Section: Two Variables: the Equation Of A Vibrating Stringmentioning

confidence: 99%

Rounding Error Analysis of Linear Recurrences Using Generating Series

Mezzarobba

2020

Preprint

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We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of the approach by several nontrivial application examples. Among these examples are a new worst-case analysis of an algorithm for computing Bernoulli numbers, and a new algorithm for evaluating differentially finite functions in interval arithmetic while avoiding interval blow-up.

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“…The D-finiteness of generating functions reflects the complexity of combinatorial classes [Pak18]. Since this class is closed under addition, multiplication, and the process of taking diagonals, it has become a useful data structure for the manipulation of special functions in symbolic computation [Sal17]. The main goal of this paper is to study D-finite power series in the framework of rational dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Rational dynamical systems, $S$-units, and $D$-finite power series

Bell,

Chen,

Hossain

2020

Preprint

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Let K be an algebraically closed field of characteristic zero and let G be a finitely generated subgroup of the multiplicative group of K. We consider K-valued sequences of the form a n := f (ϕ n (x 0 )), where ϕ : X X and f : X P 1 are rational maps defined over K and x 0 ∈ X is a point whose forward orbit avoids the indeterminacy loci of ϕ and f . Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the set of n for which a n ∈ G is a finite union of arithmetic progressions along with a set of Banach density zero. In addition, we show that if a n ∈ G for every n and X is irreducible and the ϕ orbit of x is Zariski dense in X then there are a multiplicative torus G d m and maps Ψ :We then obtain results about the coefficients of D-finite power series using these facts. Contents 1. Introduction 1 2. Linear recurrences in abelian groups 5 3. Multiplicative dependence and S-unit equations 11 4. Proof of Theorem 1.1 17 5. Heights of points in orbits 20 6. Applications to D-finite power series 23 References 27

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Linear Differential Equations as a Data Structure

Cited by 19 publications

References 108 publications

Stability Problems in Symbolic Integration

Stability Problems in Symbolic Integration

Rounding Error Analysis of Linear Recurrences Using Generating Series

Rational dynamical systems, $S$-units, and $D$-finite power series

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